Armature doesn't stay connected when moving mesh in object mode
Armature Deformation usually works from bone to mesh, not the other way around. There is an option to use a shape key to deform the mesh, and then parent a bone to some of the vertices, but it will not be applicable to your situation. Depending on the rig itself, there might however be a workaround.

Given that the bones have to deform the mesh, what you need to achieve then is to reposition all of your bones in Pose Mode so they form a circle. To do this, you can try the following: Set the rotation mode to individual origins with the button on the bottom left of the screenshot:



This will apply the rotation to each child of the bone individually, instead of just rotating the upmost parent. The bone chain in my screenshot is one single chain, so all bones are in a parent/child relationship. If you now select all bones, and then activate the rotation tool, constraining the rotation to Z-Axis only, you should be able to achieve this:



Rotate it until the circle closes and you're done. In Edit Mode the chain will still be straight, also if you disable the armature modifier visibility you're back to a straight chain.

The logic is: The first bone rotates by X degrees, its first child by X degrees again, the child of that child by another X degrees, and so on. That's why you get a circle in the end. This method is sometimes used when animating fingers curling. If that works with your current rig has to be tested, but I believe there is a chance it will.